31 problems

Let $a\colon {\mathbb N}\to {\mathbb Z}$ be a sequence. A sequence is good for $\ell$-step irrational equidistribution if it satisfies the corresponding irrational equidistribution…

Restricted density polynomial Hales–Jewett conjecture. There exists $N=N(d,\delta)\in\mathbb N$ such that, for every $S\subseteq\mathcal P(\{1,\ldots,N\}^d)$ satisfying

VIP-system recurrence conjecture. The VIP-system $v$ is good for nice recurrence.

Let $G$ be a group. A set $R\subset G$ is a set of measurable recurrence if, for every measure-preserving $G$-system $(X,\mathcal{X},\mu,(T_g))$ and every $A\in\mathcal{X}$ with…

Brown–Graham–Landman conjecture. If $S$ is not a set of $d$-nilBohr recurrence for some $d$, then $S$ is not $2$-large.

Katznelson's conjecture. There is a set $S\subset\mathbb{N}$ which is a set of Bohr recurrence but not a set of topological recurrence.

Glasner–Huang–Shao–Weiss–Ye conjecture. The set $R$ has nonempty intersection with every infinite arithmetic progression of step size $k$.

Let $G$ be a virtually-$\mathbb{Z}$ group and let $H$ be a nontrivial finite group. Let $\lambda_c$ be the critical parameter such that the $\lambda$-stationary random walk on…

Unbounded-jump random-walk conjecture. In this setup, the local version of the limit law for $V_n(1)$ is also true, and the random walk defined above is recurrent.

Let $(X,T)$ and $(X,S)$ be minimal systems on the same compact space $X$. Mixed linear-quadratic nonrecurrence conjecture. There are minimal systems $(X,T)$ and $(X,S)$ such that f…

A topological dynamical system is a pair $(X,T)$ consisting of a compact space $X$ and a continuous map $T:X\to X$; it is minimal if every orbit is dense. For two transformations…

Let $G$ be a finite abelian group of order $N$, let $\textup{End}(G)$ denote its endomorphism ring, and let $A\subseteq G$. For $\alpha,\varepsilon>0$, write…

For $r\in\mathbb N$, let $BH:=BH(\bm\beta,\bm y,k,\varepsilon)$ be a proper Bohr-Hamming Ball, with $\varepsilon>0$ and $y\in\mathbb T^r$. Define … A set is $(\delta,k)$-recurrent…

Popular-difference density conjecture. The displayed lower bound holds for every pair of matrices satisfying the stated full-rank condition. The conjecture is a finitary form of th…

Katznelson's finite-field conjecture. Every subset of $\mathbb F_p^\omega$ which is a set of Bohr recurrence is a set of chromatic recurrence.

Special Katznelson conjecture. If $S\subseteq\Delta_2(\mathcal E_1)$ is a set of Bohr recurrence in $\mathbb Z^\omega$, then $S$ is a set of chromatic recurrence.

Katznelson's conjecture. If $S\subseteq G$ is a set of Bohr recurrence, then $S$ is a set of chromatic recurrence.

Let $S\subseteq\mathbb Z$ be a set of chromatic recurrence, let $k\in\mathbb N$, let $C>0$, and let $\widetilde H(\boldsymbol\alpha;C\sqrt d,\varepsilon)$ denote the set defined ea…

Let $S\subseteq\mathbb Z$ be a set of density recurrence, meaning that for every subset $A$ of a suitable positive upper density, $(A-A)\cap S\neq\varnothing$. Hereditary recurrenc…

Let $\boldsymbol{\kappa}_\lambda$ be the left eigenvector associated with the dual Markov chains, and let $\lambda$ be the parameter used to construct them. The chains are called t…

Exponential-mixing conjecture. If $f$ is exponentially mixing, then the conclusion of Proposition (a) holds for all non-periodic points $x$.

$p$-rotor recurrence conjecture. For almost every such $\rho$, the corresponding $p$-rotor walk on $\mathbb{Z}^2$ visits every vertex infinitely many times almost surely.

Let $T$ be a triangulation, let $x$ be a tangent vector on $T$, and let $G^x$ denote the corresponding graph of discrete tangent vectors. Suppose that $G^x$ is recurrent and that t…

Let $S\subseteq\mathbb Z\setminus\{0\}$ be a set of recurrence. A Borel probability measure $\sigma$ on $\mathbb T$ is continuous if $\sigma(\{a\})=0$ for every $a\in\mathbb T$, an…

Let $X$ be a one-dimensional random-walk increment with zero mean, and let $G_n$ denote the centre of mass at time $n$. Recurrence conjecture. Suppose $d=1$. If … then $G_n$ is rec…